MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe1tmmul2 Structured version   Visualization version   GIF version

Theorem coe1tmmul2 22279
Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z 0 = (0g𝑅)
coe1tm.k 𝐾 = (Base‘𝑅)
coe1tm.p 𝑃 = (Poly1𝑅)
coe1tm.x 𝑋 = (var1𝑅)
coe1tm.m · = ( ·𝑠𝑃)
coe1tm.n 𝑁 = (mulGrp‘𝑃)
coe1tm.e = (.g𝑁)
coe1tmmul.b 𝐵 = (Base‘𝑃)
coe1tmmul.t = (.r𝑃)
coe1tmmul.u × = (.r𝑅)
coe1tmmul.a (𝜑𝐴𝐵)
coe1tmmul.r (𝜑𝑅 ∈ Ring)
coe1tmmul.c (𝜑𝐶𝐾)
coe1tmmul.d (𝜑𝐷 ∈ ℕ0)
Assertion
Ref Expression
coe1tmmul2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥,   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥,
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 coe1tmmul.r . . 3 (𝜑𝑅 ∈ Ring)
2 coe1tmmul.a . . 3 (𝜑𝐴𝐵)
3 coe1tmmul.c . . . 4 (𝜑𝐶𝐾)
4 coe1tmmul.d . . . 4 (𝜑𝐷 ∈ ℕ0)
5 coe1tm.k . . . . 5 𝐾 = (Base‘𝑅)
6 coe1tm.p . . . . 5 𝑃 = (Poly1𝑅)
7 coe1tm.x . . . . 5 𝑋 = (var1𝑅)
8 coe1tm.m . . . . 5 · = ( ·𝑠𝑃)
9 coe1tm.n . . . . 5 𝑁 = (mulGrp‘𝑃)
10 coe1tm.e . . . . 5 = (.g𝑁)
11 coe1tmmul.b . . . . 5 𝐵 = (Base‘𝑃)
125, 6, 7, 8, 9, 10, 11ply1tmcl 22275 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
131, 3, 4, 12syl3anc 1373 . . 3 (𝜑 → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
14 coe1tmmul.t . . . 4 = (.r𝑃)
15 coe1tmmul.u . . . 4 × = (.r𝑅)
166, 14, 15, 11coe1mul 22273 . . 3 ((𝑅 ∈ Ring ∧ 𝐴𝐵 ∧ (𝐶 · (𝐷 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
171, 2, 13, 16syl3anc 1373 . 2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
18 eqeq2 2749 . . . 4 ((((coe1𝐴)‘(𝑥𝐷)) × 𝐶) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
19 eqeq2 2749 . . . 4 ( 0 = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
20 coe1tm.z . . . . . . 7 0 = (0g𝑅)
211adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Ring)
22 ringmnd 20240 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Mnd)
24 ovex 7464 . . . . . . . 8 (0...𝑥) ∈ V
2524a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0...𝑥) ∈ V)
26 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷𝑥)
274adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℕ0)
28 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℕ0)
29 nn0sub 12576 . . . . . . . . . 10 ((𝐷 ∈ ℕ0𝑥 ∈ ℕ0) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3027, 28, 29syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3126, 30mpbid 232 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ ℕ0)
3227nn0ge0d 12590 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 0 ≤ 𝐷)
33 nn0re 12535 . . . . . . . . . . 11 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
3433ad2antrl 728 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℝ)
354nn0red 12588 . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
3635adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℝ)
3734, 36subge02d 11855 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0 ≤ 𝐷 ↔ (𝑥𝐷) ≤ 𝑥))
3832, 37mpbid 232 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ≤ 𝑥)
39 fznn0 13659 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4039ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4131, 38, 40mpbir2and 713 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ (0...𝑥))
421ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43 eqid 2737 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
4443, 11, 6, 5coe1f 22213 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
452, 44syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4645ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1𝐴):ℕ0𝐾)
47 elfznn0 13660 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
4847adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
4946, 48ffvelcdmd 7105 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
50 eqid 2737 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
5150, 11, 6, 5coe1f 22213 . . . . . . . . . . . 12 ((𝐶 · (𝐷 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5213, 51syl 17 . . . . . . . . . . 11 (𝜑 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5352ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
54 fznn0sub 13596 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
5554adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥𝑦) ∈ ℕ0)
5653, 55ffvelcdmd 7105 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾)
575, 15ringcl 20247 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5842, 49, 56, 57syl3anc 1373 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5958fmpttd 7135 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
601ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝑅 ∈ Ring)
613ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐶𝐾)
624ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ∈ ℕ0)
63 eldifi 4131 . . . . . . . . . . . . 13 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → 𝑦 ∈ (0...𝑥))
6463, 54syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → (𝑥𝑦) ∈ ℕ0)
6564adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (𝑥𝑦) ∈ ℕ0)
66 eldifsn 4786 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷)))
67 simplrl 777 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
6867nn0cnd 12589 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
6947nn0cnd 12589 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
7069adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
7168, 70nncand 11625 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥𝑦)) = 𝑦)
7271eqcomd 2743 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥𝑦)))
73 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝐷 = (𝑥𝑦) → (𝑥𝐷) = (𝑥 − (𝑥𝑦)))
7473eqeq2d 2748 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥𝑦) → (𝑦 = (𝑥𝐷) ↔ 𝑦 = (𝑥 − (𝑥𝑦))))
7572, 74syl5ibrcom 247 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥𝑦) → 𝑦 = (𝑥𝐷)))
7675necon3d 2961 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥𝐷) → 𝐷 ≠ (𝑥𝑦)))
7776impr 454 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷))) → 𝐷 ≠ (𝑥𝑦))
7866, 77sylan2b 594 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ≠ (𝑥𝑦))
7920, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78coe1tmfv2 22278 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
8079oveq2d 7447 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
815, 15, 20ringrz 20291 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8242, 49, 81syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8363, 82sylan2 593 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8480, 83eqtrd 2777 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
8584, 25suppss2 8225 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) supp 0 ) ⊆ {(𝑥𝐷)})
865, 20, 23, 25, 41, 59, 85gsumpt 19980 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)))
87 fveq2 6906 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1𝐴)‘𝑦) = ((coe1𝐴)‘(𝑥𝐷)))
88 oveq2 7439 . . . . . . . . . 10 (𝑦 = (𝑥𝐷) → (𝑥𝑦) = (𝑥 − (𝑥𝐷)))
8988fveq2d 6910 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))))
9087, 89oveq12d 7449 . . . . . . . 8 (𝑦 = (𝑥𝐷) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
91 eqid 2737 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))
92 ovex 7464 . . . . . . . 8 (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) ∈ V
9390, 91, 92fvmpt 7016 . . . . . . 7 ((𝑥𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9441, 93syl 17 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9528nn0cnd 12589 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℂ)
9627nn0cnd 12589 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℂ)
9795, 96nncand 11625 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥 − (𝑥𝐷)) = 𝐷)
9897fveq2d 6910 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
993adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐶𝐾)
10020, 5, 6, 7, 8, 9, 10coe1tmfv1 22277 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10121, 99, 27, 100syl3anc 1373 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10298, 101eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = 𝐶)
103102oveq2d 7447 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
10486, 94, 1033eqtrd 2781 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
105104anassrs 467 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
1061ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
1073ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐶𝐾)
1084ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
10954ad2antll 729 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℕ0)
11054nn0red 12588 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℝ)
111110ad2antll 729 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℝ)
11233ad2antlr 727 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
11335ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
11447ad2antll 729 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115114nn0ge0d 12590 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
11647nn0red 12588 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117116ad2antll 729 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118112, 117subge02d 11855 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥𝑦) ≤ 𝑥))
119115, 118mpbid 232 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ≤ 𝑥)
120 simprl 771 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ¬ 𝐷𝑥)
121112, 113ltnled 11408 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷𝑥))
122120, 121mpbird 257 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123111, 112, 113, 119, 122lelttrd 11419 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) < 𝐷)
124111, 123gtned 11396 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥𝑦))
12520, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124coe1tmfv2 22278 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
126125oveq2d 7447 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
12745ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (coe1𝐴):ℕ0𝐾)
128127, 114ffvelcdmd 7105 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
129106, 128, 81syl2anc 584 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
130126, 129eqtrd 2777 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
131130anassrs 467 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
132131mpteq2dva 5242 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133132oveq2d 7447 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
1341, 22syl 17 . . . . . . 7 (𝜑𝑅 ∈ Mnd)
13520gsumz 18849 . . . . . . 7 ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136134, 24, 135sylancl 586 . . . . . 6 (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137136ad2antrr 726 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138133, 137eqtrd 2777 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 )
13918, 19, 105, 138ifbothda 4564 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ))
140139mpteq2dva 5242 . 2 (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
14117, 140eqtrd 2777 1 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  ifcif 4525  {csn 4626   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155   < clt 11295  cle 11296  cmin 11492  0cn0 12526  ...cfz 13547  Basecbs 17247  .rcmulr 17298   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  .gcmg 19085  mulGrpcmgp 20137  Ringcrg 20230  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184
This theorem is referenced by:  coe1tmmul2fv  22281  coe1sclmul2  22287
  Copyright terms: Public domain W3C validator